Local-global principles make it possible to understand the properties of global objects in algebra or geometry by analyzing the objects locally. The study of local-global principles for the existence of rational points on homogeneous spaces under linear algebraic groups over number fields has a long history; the Hasse-Minkowski theorem that quadrics which admit local points at all places of the number field admit rational points is one of the earliest in this direction. A similar study over function fields of curves over p-adic fields has led to a host of results parallel to the number field case. In this talk, we shall trace recent developments in this direction.